Molecular Nature of S: Thermal Entropy (uakron.edu, 20min) We can explain configurational entropy by studying particles in boxes, but only at constant temperature. How does the entropy change if we change the temperature? Why should it change if we change the temperature? The key is to recognize that energy is quantized, as best exemplified in the Einstein Solid model. We learned in Chapter 1 that energy increases when temperature increases. If we have a constant number of particles confined to lattice locations, then the only way for the energy to increase is if some of the molecules are in higher energy states. These "higher energy states" correspond to faster (higher frequency) vibrations that stretch the bonds (Hookean springs) to larger amplitudes. We can count the number of molecules in each energy state similar to the way we counted the number of molecules in boxes. Then we supplement the formula for configurational entropy changes to arrive at the following simple relation for all changes in entropy for ideal gases: ΔS = Cv ln(T₂/T₁) + R ln(V₂/V₁). Note that we have related the entropy to changes in state variables. This observation has two significant implications: (1) entropy must also be a state function (2) we can characterize the entropy by specifying any two variables. For example, substituting V = RT/P into the above equation leads to: ΔS = Cp ln(T₂/T₁) - R ln(P₂/P₁).

Comprehension Questions:
1. Show the steps required to derive ΔS = Cp ln(T₂/T₁) - R ln(P₂/P₁) from ΔS = Cv ln(T₂/T₁) + R ln(V₂/V₁).
2. We derived a memorable equation for adiabatic, reversible, ideal gases in Chapter 2. Hopefully, you have memorized it by now! Apply this formula to compute the change in entropy for adiabatic, reversible, ideal gases as they go through any change in temperature and pressure.
3. Make a table enumerating all the possibilities for 3 oscillators with 4 units of energy. 
4. Compute the change in entropy (J/k) for 100 oscillators going from 3 units of energy to 50 units of energy.
5. Compute the change in entropy (J/K) for 100 particles going from 3 boxes to 50 boxes. (This is a review of configurational entropy.)

Residue Curve Modeling using Matlab/chap16/residue.m (8:00) (msu.edu)

Residue curves are powerful guides for distillation column design. Residue curves can be generated using bubble temperature calculations as described in the textbook. This screencast describes the strategy to generate a residue map by generating a series of curves and then inferring the location of the separatrices (distillation boundaries).

Hints for Generating LLE Envelopes (2:25) (msu.edu)

This screencasts makes several recommendations that help generate LLE phase envelopes most successfully.

This example hypothesizes a "pre-quel" to Example 10.1 in the form of a liquid reactor at 20 bars and asks what temperature the reactor must have been in order to result in the flash at 320K and 8 bars if no heat was added. This requires an adiabatic flash calculation. (7min, uakron.edu) The procedure demonstrated here applies the enthalpy pathway of Fig. 2.6c, with Eqn. 2.45 to estimate heats of vaporization. With this approach, you should be able to solve for mass and energy balances of any mixture at any vapor fraction. You should watch the video about Multicomponent VLE for Ideal Solutions before this one (see link above).

Note: This is a companion file in a series. You may wish to choose your own order for viewing them. For example, you should implement the first three videos before implementing this one. Also, you might like to see how to quickly visualize the Txy analog of the Pxy phase diagram. If you see a phase diagram like the ones in section 11.8, you might want to learn about LLE phase diagrams. The links on the software tutorial present a summary of the techniques to be implemented throughout Unit3 in a quick access format that is more compact than what is presented elsewhere. Some students may find it helpful to refer to this compact list when they find themselves "not being able to find the forest because of all the trees."

Comprehension Questions
1. Make a spreadsheet like the one in the video. Modify the compositions  to make a binary system like Example 10.2. Can you reproduce the results of Example 10.2?
2. Suppose a reactor was at 380K and 2MPa with a composition of {0.115, 0.335, 0.15, 0.15, 0.25} for {propane, isobutane, nbutane, isopentane, npentane}. What would be the adiabatic T&q of this stream exiting a valve at 8 bars?